Although the viscoelastic effect is linear, the system of PDEs we obtain is nonlinear due to the fiber inextensibility constraint. For a planar filament, we prove well-posedness of the fiber evolution problem, including global existence and uniqueness for small data and local existence and uniqueness for large data. The model inherits many of the features of the Newtonian problem, studied in detail from a PDE perspective in our previous work [1]}, but is fundamentally different in that the analysis now includes an additional ODE for the memory variable. As in [1]}, we prove that given a (small) time-periodic forcing along the fiber in the form of a preferred curvature, there exists a unique periodic solution to the filament evolution equations. Furthermore, we show that the periodic solution to the viscoelastic PDE converges to the unique periodic solution in the Newtonian setting as a parameter corresponding to the fluid relaxation time goes to zero. These analysis questions are interesting in their own right and continue to develop the PDE theory of the hydrodynamics of slender filaments initiated in [3]}, [4]}, [5]}, [6]}, [7]}.
